Optimal resource distribution program in a two-sector economic model with a Cobb-Douglas production function with distinct amortization factors (Q1943718)

The authors consider the following two-dimensional optimal control problem on a fixed time interval: \[ \begin{align*}{\dot{x}_1 & = \frac{u}{\varepsilon_1}F(x) - \mu_1x_1, \qquad x_1(0) = x_{10} > 0, \cr \dot{x}_2 & = \frac{1-u}{\varepsilon_2}F(x) - \mu_2x_2, \quad x_2(0) = x_{10} > 0, \cr J & \equiv x_2(T) \longrightarrow \max, \qquad 0 \leq u \leq 1, \cr}\end{align*} \] where \(x_1\) and \(x_2\) are positive state variables, \(u\) is a control, and \(F(x) = x_1^{\varepsilon_1}x_2^{\varepsilon_2}\) is the Cobb-Douglas type function. \(T\) is the given planning horizon. By applying the maximum principle, the authors prove the optimality of an extremal solution.